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## Precalculus

### Unit 8: Lesson 7

Theoretical & empirical probability distributions- Theoretical probability distribution example: tables
- Theoretical probability distribution example: multiplication
- Develop probability distributions: Theoretical probabilities
- Probability distributions from empirical data
- Develop probability distributions: Empirical probabilities

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# Theoretical probability distribution example: tables

AP.STATS:

VAR‑5 (EU)

, VAR‑5.A (LO)

, VAR‑5.A.1 (EK)

, VAR‑5.A.2 (EK)

, VAR‑5.A.3 (EK)

CCSS.Math: We can create the probability distribution for rolling two dice by creating a table to represent the events that make up the sample space. Created by Sal Khan.

## Video transcript

- [Instructor] We're
told that a board game has players roll two
3-sided dice, these exist and actually I looked it up, they do exist and they're actually fascinating. And subtract the numbers
showing on the faces. The game only looks at
non-negative differences. For example, if a player
rolls a one and a three, the difference is two. Let D represent the
difference in a given roll. Construct the theoretical
probability distribution of D. So pause this video and see
if you can have a go at that before we work through it together. All right, now let's
work through it together. So let's just think about
all of the scenarios for the two die. So let me draw a little table here. So let me do it like that and let me do it like this. And then let me put a little
divider right over here. And for this top, this
is going to be die one and then this is going to be die two. Die one can take on one, two, or three and die two could be one, two, or three. And so let me finish making
this a bit of a table here. And what we wanna do is
look at the difference but the non-negative difference. So we'll always subtract the
lower die from the higher die. So what's the difference here? Well, this is going to be zero. If I roll a one and a one. Now, what if I roll a two and a one? Well, here the difference is
going to be two minus one, which is one. Here the difference is three
minus one, which is two. Now what about right over here? Well, here the higher die
is two the lower one is one, right over here. So two minus one is one, two minus two is zero. And now this is gonna be the higher roll, die one is gonna have the
high roll in this scenario. Three minus two is one. And then right over here,
three minus one is two. Now die one rolls a two, die two rolls a three. Die three is higher,
three minus two is one. And then three minus three is zero. So we've come up with all of the scenarios and we can see that we're
either gonna end up with a zero or one or a two when we look
at the positive difference. So there's a scenario of
getting a zero, a one or a two. Those are the different differences that we could actually get. And so let's think about the
probability of each of them. What's the probability that
the difference is zero. Well, we can see that one, two, three of the nine equally likely outcomes, result in a difference of zero. So it's gonna be three
out of nine or one-third. What about a difference of,
let me use the blue, one? Well, we could see there
are one, two, three, four of the nine scenarios have that. So there is a four ninths probability. And then last but not
least a difference of two. Well, there's two out of the
nine scenarios that have that. So there is a two ninths
probability right over there. And we're done. We've constructed the theoretical
probability distribution of D.